The Mack-Method and Analysis of
Variability
Prepared by Erasmus Gerigk
Presented to the Institute of Actuaries of Australia
Accident Compensation Seminar 28 November to 1 December 2004.
This paper has been prepared for the Institute of Actuaries of Australia’s (IAAust) Accident Compensation Seminar,
2004. The IAAust Council wishes it to be understood that opinions put forward herein are not necessarily those of
the IAAust and the Council is not responsible for those opinions.
© 2004 Institute of Actuaries of Australia
The Institute of Actuaries of Australia
Level 7 Challis House 4 Martin Place
Sydney NSW Australia 2000
Telephone: +61 2 9233 3466 Facsimile: +61 2 9233 3446
Email: insact@actuaries.asn.au Website: www.actuaries.asn.au
1
INTRODUCTION .............................................................................................3
SECTION 1: PRELIMINARIES........................................................................5
1.1 REVIEW OF THE THEORY OF POINT ESTIMATION ............................................5
1.2 PROCESS ERROR AND ESTIMATION ERROR .................................................5
1.3 TRIANGULATED DATA AND NOTATION ...........................................................6
SECTION 2: THE INCREMENTAL LOSS RATIO METHOD ..........................8
2.1 FORMULATION OF THE MODEL ....................................................................8
2.2 THE ESTIMATORS OF THE MODEL PARAMETERS ............................................9
2.3 ANALYSIS OF VARIABILITY ........................................................................11
SECTION 3: THE CHAIN-LADDER METHOD..............................................14
3.1 FORMULATION OF THE MODEL ..................................................................14
3.2 THE ESTIMATORS OF THE MODEL PARAMETERS ..........................................16
3.3 ANALYSIS OF VARIABILITY ........................................................................18
SECTION 4: CALCULATION OF A RISK MARGIN .....................................21
SECTION 5: TESTING THE MODEL ASSUMPTIONS .................................23
5.1 TESTING THE ADDITIVE MODEL ................................................................23
5.2 TESTING THE MULTIPLICATIVE MODEL ......................................................24
5.3 CHECKING FOR CALENDAR-YEAR EFFECTS ................................................25
SECTION 6: PRACTICAL CONSIDERATIONS............................................27
6.1 DEALING WITH OUTLIERS..........................................................................27
6.2 DECIDING BETWEEN THE MULTIPLICATIVE AND ADDITIVE MODEL ...................28
6.3 REDUCTION OF PARAMETERS ...................................................................31
APPENDIX A: FORMULAS FOR THE INCREMENTAL LOSS RATIO
METHOD .......................................................................................................33
APPENDIX B: FORMULAS FOR THE CHAIN-LADDER METHOD .............34
BIBLIOGRAPHY ...........................................................................................35
2
Introduction
The Background
This paper aims to contribute to the current discussion of estimating central
estimates and risk margins under APRA’s GPS 210. Two statistical reserving
methods are presented in detail, namely the “Incremental Loss Ratio Method”
and the “Chain-Ladder Method”.
Both methods are in fact as old as loss-reserving itself, and the underlying
formulas constitute a widely accepted heuristic method of computing IBNR
estimates - yet these methods are sometimes not further justified in terms of
mathematical statistics.
It was Thomas Mack’s achievement in the nineties to interpret both methods
within a rigorous statistical framework and to extend these methods to include
variability estimates (see Mack [1993, 1994, 1997]). It has now become
common practice to refer to this interpretation and extension of the ChainLadder-Method as the “Mack-Method” and this article extends this notion to
the Incremental Loss Ratio Method, which is not as well-known in Australia,
mainly because the only publication so far is written in German (Mack [1997,
2002]).
The key features of both methods in the context of risk margins may be
summarised as follows:
•
•
•
The variability estimates depend only on the insurers own data
The underlying assumptions can be tested
The methods are simpler than most other methods,
“Bootstrapping” or sophisticated stochastic modelling
e.g.
The biggest caveat is probably that in many cases the data provides a bad fit
to the proposed models and therefore the results need to be carefully
reviewed.
This paper
This paper intends to give a statistically sound overview of the Incremental
Loss Ratio Method and the Chain-Ladder Method without any formal proofs.
The text does not reveal anything genuinely new to the expert and it is not its
purpose to add anything to current research, although the Incremental Loss
Ratio Method with variability estimates might be unknown to many actuaries.
This was the original motivation for presenting this paper in Australia. The
method is fairly simple from a statistical point of view at least compared to the
Chain-Ladder method.
The whole Chain-Ladder theory has already been developed in Mack [1993]
and has been further explored since then at a sophisticated level in many
articles. This article gives an introduction to the practitioner into the more
difficult Chain-Ladder theory.
3
The example spreadsheets
Instead of adding lengthy numerical examples to the text, this text is
accompanied by two spreadsheets, one for either method, which demonstrate
how the methods may be implemented. They reader may find all formulas
from sections 2-4 there, only the sections on model testing and regression
have been left out. The spreadsheets shall serve two purposes: first, to
provide a better understanding of the underlying methods. Second, they
serve to support the claim that the methods are indeed simple. They were
designed for study-purposes only and should not be used if anything depends
on the correct result of the calculations.
The structure of this article
Sections 1-3 develop the basic theory. The focus lies on motivating the
crucial concepts, such as the role of model assumptions and not so much in
providing complete technical proofs without becoming too vague or imprecise.
Section 4 briefly discusses a popular method of translating variability
estimates into risk margins.
Section 5 sketches how the model assumptions for both methods can be
tested.
Section 6 addresses some additional issues that arise in practice.
4
SECTION 1: PRELIMINARIES
1.1 Review of the theory of point estimation
Although we may assume that the reader is familiar with the basic concepts of
statistical inference, it is helpful to recall some concepts and precise
terminology from the theory of point estimation.
The problem of point estimation can loosely be described as follows: assume
that the characteristics of the elements of a population can be represented by
some random vector X with a certain unknown density fx. Further assume that
the values (x1,…,xn) of the random sample X1, … , Xn can be observed. On the
basis of the observed (x1,…,xn) it is desired to estimate some function, say
t=t(fx), of the underlying distribution. In our case, the primary focus consists of
estimating the mean and the variance of the loss distributions for a particular
underwriting year or a total portfolio.
In general, point estimation admits two problems: the first, to devise some
means of obtaining estimators T(x1,…xn) for t. The second is to select criteria
to define and find a “best” estimator among many possible estimators. The
criterion selected for our purposes will be the mean-squared error (mse). It
is a popular and useful criterion, though it is perhaps crude. In many cases in
applied statistics, the mse can be calculated by making suitable distribution
assumptions about the underlying “true” X. It is an intriguing result of the
Mack-Method that we will be in a position to estimate the mse of our
estimators without the need to specify any distributions.
Instead of specifying or parameterising any distributions, our proposed
assumptions will only model how the distributions are affected by changes in
exposure or volume. This in connection with some independence
assumptions will suffice to arrive at estimators and optimality conclusions.
Note that in order to apply the theory of point estimation properly, it is
necessary to state explicitly some model assumptions about the nature of
the random vector X. It is probably this point that is sometimes lost sight of
when developing actuarial techniques in practice. One of the advantages of
making precise model assumptions is the fact that only then we can actually
test and possibly reject them, rather than intuitively decide on which model to
apply.
1.2 Process Error and Estimation Error
Throughout this article, we have to carefully distinguish between several kinds
of variability: Given some observed data set D suppose we have some
estimator Cˆ = Cˆ ( D ) of the outcome of a “true” but unknown random variable
C , (in the sequel C can be a loss-distribution and D is the data-triangle that
has been observed so far). In a sense that can be made mathematically
5
precise, the best predictor of C , given the data D , is the conditional
expectation E (C | D ) . We do not know E (C | D ) and thus have to estimate it.
This gives rise to two sources of variability. First, our estimate Ĉ could be far
away from the true E (C | D ) . Second, even if we were lucky enough to
exactly determine E (C | D ) , the prediction of the individual outcome C would
still be subject to the uncertainty Var (C | D ) . In order to make this important
observation more precise, let us introduce some terminology that will be
useful in the subsections to follow:
The Estimation Error measures the variability of the Cˆ around the “true
mean value” of the distribution we are trying to estimate. It is defined by
E[Cˆ − E (C | D )] 2 . This component is subject to our choice of estimators.
The Process Error measures the variability of the random variable C . It is
defined by Var (C | D) = E{[ E (C | D) − C ] 2 | D} . As should be intuitively clear,
there is nothing we can do to reduce the Process Error, which is inherent to
the process we are trying to forecast.
The Prediction Error measures the variability of the deviations C − Cˆ . It is
defined by mse(Cˆ | D ) = E (C − Cˆ ) 2 | D .
(
)
Looking at the identity C − Cˆ = C − E (C | D ) + E (C | D ) − Cˆ we see that the
Prediction Error is made up of two components. Indeed, after some
calculations it can be shown that
mse(Cˆ | D ) = Var (C | D ) + [ E (C | D ) − Cˆ ] 2
Thus the important result
Prediction Error = Process Error + Estimation Error
also holds when working with conditional variances.
The reader may recall the above discussion from the theory of simple linear
regression. When forecasting an individual response, the error of the
projection can be split into the Estimation and Process Error. It is an
interesting feature of the Mack-Method that it also permits an explicit split of
the Prediction Error into its two components. This can give the actuary some
insight where the resulting variability of his estimates actually stems from.
1.3 Triangulated data and notation
We may assume that the reader is familiar with the basics of run-off triangles.
Most of the notation used in the sequel should be intuitively clear. Yet to
avoid ambiguity, we give an overview of notations in order to give a formal
representation of triangulated data.
6
Overview of basic notation:
i
k
vi
S i ,k
accident-year or underwriting year period
development year
the exposure measure for underwriting year i
incremental payments made in underwriting year i and development
C i ,k
year k
cumulative payments made in underwriting year i and development
mi , k
year k
incremental loss ratio in underwriting year i and development year k
In most cases the exposure measure vi will represent the ultimate premium
or vehicle count of that particular underwriting year, though these are not the
only possible exposure measures.
An important remark:
For the sake of simplicity we will only refer to triangulated data of
payments throughout this article and we will refer to the rows of a run-off
triangle as underwriting years. This does hopefully not obscure the fact that
all methods introduced in the sequel can, in principle, be applied to a much
wider range of data, e.g. incurred claims, claims counts and possibly other
data and that the rows may as well represent accident years.
7
SECTION 2: THE INCREMENTAL LOSS RATIO METHOD
In this section we formulate the model assumptions for the Incremental Loss
Ratio method and discuss the implied estimators without proofs. The key
results are the estimators for the model parameters and the recursion
formulae for the standard error of the reserves. Sections 2.1 to 2.3 rely
heavily on Mack [2002].
2.1 Formulation of the model
Almost all reserving make the assumption that the underwriting years of the
run-off triangle are similar to each other. The most simplistic assumption is
probably to presume that the claims payments in each underwriting follow
exactly the same distribution. Certainly this condition will almost never be
fulfilled as the volume of business written in each year will be different and
thus the underlying loss distributions in each underwriting year cannot be
identical.
Still, an obvious step from here would be to assume that each underwriting
year differs only with respect to its volume. The influence of the volume on
the underlying distributions could be modelled in accordance with a concept
that is sometimes called the individual model. To be more precise: assume
that for each development year k there is some increment ratio m k such that
the following relation holds for the expected (incremental) claims payments in
each underwriting year i:
⎛S ⎞
(AM 1) E ⎜⎜ ik ⎟⎟ = mk
⎝ vi ⎠
We add another somewhat strong, but still intuitive and statistically testable
assumption:
(AM 2) The individual payments Sik in each underwriting/development year are
independent for all i,k.
The individual model also implies how the variance of each S i ,k is affected by
the exposure-measure. Assume that for each development year k there is
some s k2 such that
⎛S
(AM 3) Var ⎜⎜ ik
⎝ vi
⎞ s k2
⎟⎟ =
⎠ vi
The condition (AM 3) is intuitively quite appealing if we think of the exposure
measure as some form of vehicle- or policy-count. It is just a precise
formulation of the well-known fact from statistics that “smaller samples have
higher variances”.
8
Once we accept that our portfolio behaves according to the individual model,
the assumptions (AM 1), (AM 2) and (AM 3) may not seem to be too bold.
Let us rephrase what (AM 1 ) - (AM 3) say. We assume:
(AM 1) For a particular development year the expected loss ratio increment is
the same for all underwriting years
(AM 2) The payment in each cell is independent of all other payments
(AM 3) The variance of the loss ratio decreases in proportion to the size of
volume
It may occur to the reader that (AM 2) is probably the most likely assumption
to be violated in reality.
Now any estimator of m k leads immediately to an estimate of the future
payments. The outstanding payments are naturally projected via Rˆ = v ⋅ mˆ ,
ik
i
k
and the estimator of the reserve of the year i is given by Rˆ i = ∑ k Rˆ ik . It is
clear that these estimators are also unbiased.
By estimating s k2 we will better understand the variability underlying the above
projections.
2.2 The estimators of the model parameters
Combining the assumptions (AM 1) - (AM 3) one can deduce the following
result:
The estimators
n +1− k
(2.1)
mˆ k =
∑S
i =1
n +1− k
∑v
i =1
(2.2)
1
sˆ =
n−k
2
k
ik
i
n +1− k
∑
i =1
⎞
⎛S
vi ⎜⎜ ik − mˆ k ⎟⎟
⎠
⎝ vi
2
are unbiased estimators of m k and s k2 .
The sum in (2.1) is in fact an exposure-weighted mean of the individual
S
increment ratios mik = ik , as can be seen from rearranging (2.1) to
vi
9
(2.3)
mˆ k =
n +1− k
∑
i =1
⎛
⎞
vi
⎜
m ⎟
⎜ n +1− k v ik ⎟
⎝ ∑i =1 i
⎠
Note that since each mik is unbiased one can easily write down plenty of
unbiased estimators for m k , e.g. the arithmetic mean of the individual
increment ratios mik , but
The weighted mean as given in (2.1) has the smallest variance
among all linear unbiased estimators of mk.
We will not prove this result, although the proofs are not particularly difficult. It
is the direct consequence of a general principle in statistics, namely that when
weighting independent estimators of a random variable, the respective
weights should be inversely proportional to the corresponding variances.
For illustration purposes, assume that the “true” increment ratio for a particular
portfolio in the 2nd development year was 53%. Then the resulting triangle
could look roughly like this:
m ik =
Illustration: Incremental Loss Ratio in the 2nd dev. year
Exposure
1999
2000
2001
2002
2003
2004
7,000
5,000
6,000
6,400
8,000
9,000
1
v 2000
v 2003
2
3
50%
47%
54%
55%
55%
4
5
S ik
vi
6
m 2000 , 2
m 2003 , 2
Note two things:
1) We should expect the observed loss ratio increment in UY 2000 to
deviate more than the increment in UY 2003 from the true underlying
increment of 53%, because of the different exposures.
2) When estimating the loss ratio increment for development year 2, we
may put more credibility to the UY 2003 observation than into the UY
2000 observation.
Readers who are already familiar with the Incremental Loss Ratio Method will
immediately recognise that m̂ k is nothing but the original well-known
estimator.
10
Note that there can be no estimator for the last s k2 , because there is only one
data point in the upper right corner of the triangle and thus no estimator for
the variability of the individual incremental loss ratios. If we assume that the
data are run-off in the n-th development year, we can set sˆ n2 to 0, otherwise
we may try to interpolate from the preceding sˆ k2 , or just take their minimum as
an estimator for s k2 . This will usually lie on the safe side as one should expect
the s k2 to decline with k.
Up to here we have just reformulated what we may call the Incremental Loss
Ratio Method. Let us summarise: once we feel confident that our model
assumptions apply to our triangulated data, we are almost forced into
projecting data in accordance with the Incremental Loss Ratio Method. In
fact, we would need good reasons to diverge from that approach. This is
already something worth knowing.
We may now further investigate the
variability of the projected claims payments, which is done in the next
subsection.
2.3 Analysis of variability
The standard error of the Incremental Loss Ratio
Combining (2.3) with (AM 3) immediately tells us that the variance of the
estimator m̂ k is given by
(2.4) Var (mˆ k ) =
∑
sk2
n +1− k
i =1
vi
and this expression gives us an estimator for the s.e. of the incremental Loss
Ratios – note that because of
(2.5)
sˆk2
1
=
n +1− k
∑i=1 vi n − k
n +1− k
∑
i =1
⎛ S ik
⎞
⎜⎜
− mˆ k ⎟⎟
n +1− k
⎠
∑i =1 vi ⎝ vi
vi
2
the expression in (2.5) offers an intuitive interpretation as a “weighted
variance” of the deviations of the individual increment ratios around their
weighted mean. At this stage, it is often helpful look at the triangle of the
individual incremental loss ratios m i , k in order to better understand where the
measured variability comes from.
The standard error for a single underwriting year
We now show how the standard error of the projected reserves can be
calculated. We keep the split between random and Estimation Error.
For the practitioner, the total recursions (2.8) and (2.9) are probably the most
important.
11
The formulas presented below can be derived by direct algebraic
manipulations and do not require any higher theorems from statistics. Yet
some calculations are quite lengthy, which is why we omit them.
The Process Error for a single underwriting year is given by:
(2.6) Var (Cˆ i ,k +1 ) = vi ⋅ sˆk2+1 + Var (Cˆ i ,k )
This equation can be made intuitively clear without a formal proof:
assumption (AM 3) stated that each individual payment S i ,k has variance
v i ⋅ s k2 . Thus the variance of each cumulative Cˆ i , k +1 = Sˆ i ,k +1 + Cˆ i , k is estimated
by (2.6) since the variances just add up using the independence assumption
(AM 2).
The Estimation Error for a single underwriting year can be estimated
recursively via:
(
)
(
2
2
(2.7) Cˆ i ,k +1 − E (C i ,k +1 ) = vi2 ⋅ s.e.(mˆ k +1 ) + Cˆ i ,k − E (C i ,k )
)
2
We may confirm this equation intuitively with a bit more effort as follows: The
term s.e.(mˆ k ) measures the variation of the average estimated incremental
Loss Ratio m̂ k from the “true” incremental Loss Ratio m k , i.e. the Estimation
Error. Thus (2.7) says that the Estimation Error we are making is the
Estimation Error of the estimate Sˆi ,k plus the Estimation Error we have made
so far.
More formally, we may rewrite the Estimation Error as
2
2
Cˆ i ,k +1 − E (C i ,k +1 ) = [ Sˆ i , k − E ( S i ,k )] + [Cˆ i , k − E (C i ,k )] and then square out the
(
) (
)
sum and take expectations on both sides. This leads then to (2.7).
Putting (2.6) and (2.7) together we get immediately the following
Recursion for the standard error of a single underwriting-year:
(
(2.8) s.e. Cˆ i ,k +1
)
2
( )
2
= vi ⋅ sˆk2+1 + vi2 ⋅ s.e.(mˆ k +1 ) + s.e. Cˆ i ,k
2
The total standard error
The manipulations in the previous apply also for the total standard error – it is
basically the same reasoning only with a different volume measure. With the
n
intermediate term u k = ∑i = n− k + 2 vi , which can be thought of as the total
12
exposure corresponding to development period k+1, the following recursion
n
formula for the total standard error of Cˆ k = ∑i =1 Ci ,k holds:
Recursion for the total standard error:
( )
(2.9) s.e. Cˆ k +1
2
( )
2
= u k ⋅ sˆk2+1 + u k2 ⋅ s.e.(mˆ k +1 ) + s.e. Cˆ k
( )
with the start value s.e. Cˆ 2
2
2
( )
2
= s.e. Cˆ n , 2 . In the same way we may calculate
the Process Error and Estimation Error, just by replacing the volumes vi by
u k in formulas (2.6) and (2.7).
13
SECTION 3: THE CHAIN-LADDER METHOD
In this section we formulate the model assumptions for the Chain-Ladder and
discuss the implied estimators. The key results are the estimators for the
model parameters and the recursion formulae for the standard error of the
reserves. This section is basically a very short summary of Mack [1993] and
Mack [1997].
3.1 Formulation of the model
We will now turn to what is usually called the Chain-Ladder method. In
principle, we proceed in a similar fashion as in the previous section, starting
with some (hopefully plausible) model-assumptions and ending up with a set
of estimators. But since the model assumptions may not be as easily
motivated as in the additive case, we start by taking the CL-method as given
and look into the implicit assumptions that we may have made when applying
this method. This approach was first introduced by Mack [1993].
Let us start with some general remarks:
In the additive model, the ultimate loss Cin was written in the form
(3.1) Cin = Si1 + … + Sin
with the S ik being the “additive increment” in development year k, and then
we went on looking for estimators of S ik . A natural multiplicative approach is
to start with the identity
(3.2) Cin = Ci1 Fi1 Fi2 … Fin-1
with Fik = Ci,k /Ci,k-1 being the “multiplicative increment” in development year k,
or the k-th individual development factor in year i. In a loose sense, all we
need to do is to look for estimators of the Fik.
Although the resulting formulas are quite similar to the ones developed in
section 2, more care needs to be taken when manipulating terms - some
technical difficulties will arise from the simple fact that the Expectationsoperator is additively linear, but not compatible with respect to multiplication,
since the Cik are not uncorrelated within the underwriting years, and thus in
general we would have E (C i , k +1 / C ik ) ≠ E (C i , k +1 ) / E (C ik ) . To overcome these
difficulties we will have to remind ourselves of the notion of Conditional
Expectation and the role it plays in finding optimal predictors.
Let us see if we can motivate a model-assumption by anticipating the CLmethod:
14
When projecting the ultimate loss Cin (or next year’s loss Cik) in the
underwriting period i through some set of development factors { fˆk }, we want
to apply some formulae like
(3.3)
Cˆ in = C i ,n +1−i fˆn −i ⋅ K ⋅ fˆn
or
(3.4)
Cˆ i ,k = C i ,k −1 fˆk
which do not use any of the loss-information of the older development
periods. Here we are making a strong implicit assumption! Why, we may
ask, don’t we use some form of average of the possible projections:
3.3’)
C i1 fˆ2 ⋅ K ⋅ fˆn
C fˆ ⋅ K ⋅ fˆ
i2
3
n
L
C i ,n +1−i fˆn + 2−i ⋅ K ⋅ fˆn
A plausible answer seems to be, that we implicitly assume that we cannot
improve our estimator of Cin by using the information given in the older Cik.
Stated differently, we implicitly assume that we cannot improve our estimator
of fˆk by using any of the Ci1,…, Cik-1 . The same reasoning applies to the fact
that we do not use any of the information of the preceding underwriting
periods. Translating these thoughts into the language of probability leads
directly to the following: We assume that there are factors f k such that
⎞
⎛ C
(CL 1) E ⎜⎜ ik Ci1 ,K, Cik −1 ⎟⎟ = f k
⎠
⎝ C ik −1
1 ≤ i ≤ n;
2≤k ≤n
which is the multiplicative Chain-Ladder assumption corresponding to
equation (AM 1).
The concept of conditional expectation is crucial to understanding the
assumption we are making here. Bear in mind that a conditional expectation
is a function and not just a number. So let us reword (CL 1) once more. In a
sense that can be made mathematically precise, the equation
(CL 1) E (Ci ,k Ci1 ,K, Cik −1 ) = Ci ,k −1 f k
1 ≤ i ≤ n;
15
2≤k ≤n
tells us the following: among all functions g (C i1 , K , C ik −1 ) of the known part of
the triangle, the best predictor of C i , k is just the function
g (C ik , K , C ik −1 ) = C i ,k −1 f k for some unknown parameter f k .
We are not done yet – we will also need a model assumption for the variance
that corresponds to (AM 3). Viewing C i , k as an exposure-measure in (CL 1),
it may not come as a surprise if we write down in analogy to the additive
method:
⎞
⎛ C
σ2
(CL 3) Var ⎜ i ,k C i1 , K , C ik −1 ⎟ = k
⎟ C
⎜C
i , k −1
⎠
⎝ i ,k −1
1 ≤ i ≤ n;
2≤k ≤n
Recall that by definition Var (Ci ,k | D) = E{[Ci ,k − E (Ci ,k | D)]2 | D}
It is probably even more difficult to attach an intuitive meaning to (CL 3) than
for (CL 1). Let us try, though: we re-write (CL 3) expressing it in the form:
(CL 3) Var (Ci ,k Ci1 ,K, Cik −1 ) = Ci ,k −1σ k2
1 ≤ i ≤ n;
2≤k ≤n
We can state (CL 3) in loose words as follows: among all functions
h(C i1 , K, C ik −1 ) , the best predictor of the squared distance between the
estimate in (CL 1) and C i , k is given by h(Ci1 ,K, Cik −1 ) = Ci ,k −1σ k2 for some
parameter σ k .
A justification for (CL 3) may become clearer as soon when we look at the CLestimators for f k . It is precisely assumption (CL 3) that will make the CLestimators for f k optimal.
And finally, we make the assumption
(CL 2) The underwriting years { Ci1 , …, Cin} are globally independent, i.e. the
sets { Cj1 , …, Cjn} are independent for i≠j .
Although this assumption is quite strong, it is plausible and – more importantly
– statistically testable.
3.2 The estimators of the model parameters
Starting from the model-assumptions (CL 1) – (CL 2), it turns out that similar
results like those obtained above for the additive method can be derived.
16
The estimators
n +1− k
(3.5)
fˆk =
∑C
i =1
n +1− k
∑C
i =1
(3.6)
1
σˆ =
n−k
2
k
i ,k
i , k −1
n +1− k
∑
i =1
⎛ C i ,k
⎞
C i ,k −1 ⎜⎜
− fˆk ⎟⎟
⎝ C i ,k −1
⎠
2
are unbiased estimators of mk and σk2.
Just like in the additive case, equation (3.5) can be thought of as a weighted
C i ,k
mean of the individual development factors Fi , k =
, this time the weights
C i , k −1
being the current cumulative loss C i , k . Indeed, rearranging the sum gives us
(3.7)
fˆk =
n − k +1
∑
i =1
⎞
⎛ C
i , k −1
⎟
⎜
⋅
F
i ,k ⎟
⎜ n−k C
⎠
⎝ ∑i =1 i ,k −1
and since each of the individual Fi , k is an unbiased estimator of f k , so is their
weighted mean fˆk . Again, the particular choice of weights is justified by the
following fact, which we state without proof:
The weighted mean as given in (3.5) has the smallest variance
among all linear unbiased estimators of f k .
This last observation shows why it may be justified to claim that anyone using
the Chain-Ladder method with the development factor (3.5) has implicitly
relied on the model assumptions (CL 1)-(CL 3).
The interpretation of (3.5) similar as in the additive case, only that the
exposure this time is the “cumulative loss in the preceding year”:
17
Illustration: Cumulative incurred losses up to 2nd development year
1
2
3,501
2,350
3,212
3,550
4,421
1999 C 2000 ,1 716
2000
533
2001
753
2002 C 2003 ,1 664
2003
1,036
2004
840
3
4
5
Individual development factors in the 2nd development year
1
1999
2000
2001
2002
2003
2004
F 2000
,2
F 2003
,2
2
4.89
4.41
4.26
5.34
4.27
3
4
5
We may have an eye on two things: Firstly, how volatile the individual factors
are within each development period and secondly, how much weight is
assigned to each individual factor. This check should always be done, as the
σ k2 can be quite volatile in the first or last underwriting years and may have
erratic high values. These would be carried forward in the calculation of the
standard error. In this case, it might make sense to smoothen the extreme σ k2
by means of a regression as will be outlined in the last chapter.
We may also spot trends in the development factors here, although this is
more systematically examined by looking at residuals as will be explained in
later.
Note that there can be no estimator for the last σ n2 , because there is only one
data point in the upper right corner of the triangle and thus no estimator for
the variability of development factors. We may set σ n2 to 0 if the data are runoff, otherwise we may try to interpolate through a regression from the
preceding σ k2 .
3.3 Analysis of variability
The standard error of the Development Factor
The variance of the estimator of the development factors is given by
18
( )
(3.8) Var fˆk =
∑
σ k2
n +1− k
i =1
C i ,k −1
This formula can be derived directly and will be justified once more later via
regression-analysis in section 5. The estimator in (3.6) for σ k gives us an
estimator for Var fˆ . Note that just like in the additive case, Var fˆ is
( )
( )
k
k
determined by the fluctuations of the individual Fi , k around their weighted
mean, which can be seen by inserting (3.6) into (3.8). It is therefore helpful in
practice to look at the triangle of individual development factors Fi , k in order to
understand where the variability comes from.
The standard error for a single underwriting year
The whole exercise from the additive case can be carried out in the
multiplicative case in a similar fashion. There are recursion formulae for the
Process and Estimation Error, respectively, and for the Prediction Error of the
projected loss. We try to give an intuitive understanding of the formulas
without even attempting any formal proof. This shall not conceal the fact that
the results quoted in this section are far from obvious and that the rigorous
proofs form the hardest bit of the whole theory. The interested reader may
refer to Mack [1994] and Mack (1999).
The Process Error for a single underwriting year is:
(3.9) Var (Ci ,k +1 | D) = E (Ci ,k | D)σ k2+1 + Var (Ci ,k | D) f k2+1
which allows a similar interpretation as in the additive case: the first term
contains s.e.( Fi ,k ) 2 = σ k2 / Ci ,k which measures the variation of the individual
factors Fi ,k around their “true” mean f k , i.e. the Process Error. And we add
the Process Error of the preceding period increased by the development
factor.
The Estimation Error for a single underwriting year is given by:
(
)
(
)
2
2
(3.10) Cˆ i ,k +1 − E (C i ,k +1 | D) = Cˆ i2,k s.e.( fˆk +1 ) 2 + Cˆ i ,k − E (C i ,k | D) fˆk2+1
Again, the term s.e.( fˆk ) measures the variation of the average estimated
development factor fˆ from the “true” factor f , i.e. the Estimation Error.
k
k
For the Prediction Error we thus write:
19
(3.11)
(
s.e. Cˆ i ,k +1
)
2
⎛ σˆ 2
⎞
2
= Cˆ i2,k ⎜ k +1 + s.e.( fˆk +1 ) 2 ⎟ + fˆk2+1 ⎛⎜Var (C i ,k | D) + Cˆ i ,k − E (C i ,k | D) ⎞⎟
⎜ Cˆ
⎟
⎝
⎠
⎝ i ,k
⎠
(
)
We may reformulate the above formula once more in a more simplified
notation:
Recursion for the standard error of a single underwriting-year:
(
(3.12) s.e. Cˆ i ,k +1
)
2
( )
2
= Cˆ i ,k σˆ k2+1 + Cˆ i2,k s.e.( fˆk +1 ) 2 + fˆk2+1 ⋅ s.e. Cˆ i ,k
The total standard error
With the exposure term hk = ∑i = n − k +1 Ci ,k corresponding to the development
n
year k+1 the following recursion formula for the total standard error of
n
Cˆ k = ∑i =1 Ci ,k holds for k = 2,…, n-1:
Recursion for the total standard error:
( )
(3.14) s.e. Cˆ k +1
2
( )
= hk σˆ k2+1 + hk2 ⋅ s.e.( fˆk +1 ) 2 + fˆk2+1 ⋅ s.e. Cˆ k
( )
with the start value s.e. Cˆ 2
2
2
( )
2
= s.e. Cˆ n , 2 .
The reader may note the formal similarity between the recursion formulae of
the Chain-Ladder and the Incremental Loss Ratio estimates. It is indeed
possible to design a complex set-up in a spreadsheet that does the job in both
cases, though this does not enhance transparency.
20
SECTION 4: CALCULATION OF A RISK MARGIN
Once the mean and the standard error of the underlying loss-distribution have
been estimated, it is possible to further use this information in order to
calculate a so-called risk-margin corresponding to a given level of adequacy.
It cannot be recommended to stick to a distribution-free approach at this point,
e.g. with a one-sided version of Chebyshev's inequality such as
P( X − E ( X ) ≥ α ⋅ s.e.( X )) ≤ 1 (1 + α 2 ) . Such a general inequality cannot lead to
sharp results. In practice often an additional assumption with respect to the
shape of the underlying loss-distribution is made.
A popular and simple approach consists of assuming that the total
outstanding
claims
provisions
follow
a
Log-Normal-Distribution,
2
i.e. OCP ~ log N ( µ , σ ) , where the parameters µ and σ are uniquely
determined by the mean and the coefficient of variation of the underlying
distribution. For the sake of completeness we recite the basic formulas for the
required parameters. With E(OCP) and CV denoting the mean and coefficient
of variation, respectively, we have:
σ 2 = log(1 + CV 2 )
µ = log( E (OCP )) − 12 σ 2
If we now require the level of adequacy to be x%, we first calculate the x%percentile p of the standard normal distribution. The x% percentile P for the
original loss distribution is then given by the expression
P = exp( µ + p ⋅ σ 2 )
The resulting risk margin is then often expressed as a percentage of the
central estimate.
Two comments may seem appropriate here:
1. The log-normal model is an assumption that may not be justified in
each single case and one should be aware of some peculiar features of
this distribution. In particular, the Log-Normal-Distribution implies a
maximum possible risk margin of approximately 25.5%, which is
reached by a coefficient of variation of 75% and is then declining as a
function of the CV. The pure formulaic approach as outlined above
may therefore not be suitable for highly volatile classes.
2. It is important to bear in mind that the thus derived risk margin applies
to the respective triangle as a single entity. If the underlying class of
business forms a subclass of a insurer’s portfolio then, on an overall
basis, the insurer will benefit from correlation effects. As a result, the
sum of the individual risk margins of all subclasses can be substantially
higher than the actually required risk margin on the aggregated total
21
level. In order to take those effects into account in accordance with
APRA’s GPS 210, one may discount the individual segment margin by
some estimated “diversification benefit”. This critical issue is not
further addressed in this paper.
22
SECTION 5: TESTING THE MODEL ASSUMPTIONS
In the sequel, we will introduce some means with which to test whether the
model assumptions for either the additive or the multiplicative method are
fulfilled. This should form an important step in any proper statistical analysis.
It turns out that the model-assumptions of the additive and the multiplicative
method can be interpreted as a linear regression, an area in which statistical
tests are well understood and are included in most statistical software
packages. We quote the most basic tests. A nice overview of the deeper
connections between the presented Chain-Ladder-type methods and their
interpretation as weighted regressions can be found in Murphy [1994]. This
section relies heavily on Mack [1994]. The interested reader will find much
more on testing in this article.
5.1 Testing the Additive Model
Let us recall the model assumptions for the Incremental Loss Ratio method:
(AM 1) E (S ik ) = vi m k
(AM 2) The individual payments Sik in each underwriting/development year are
independent for all i,k.
(AM 3) Var (S ik ) = vi s k2
For a fixed k, the model-assumptions can be viewed as a linear regression of
the form
(5.1) Yi = a + xi b + ε i
with a = 0 and mk = b . Looking at the error term ε i we see that because of
(5.2) Var (ε i ) = vi s k2
the variances for each observation are not constant. Such regression models
are called heteroskedastic (as opposed to the equal variance models that are
called homoskedastic). Since the parameter s k2 is unknown, but the weights
vi are known, textbook theory has it that the best linear unbiased estimator
for m k is determined via weighted least squares. It is the unique value that
minimises the expression
(5.3)
n +1− k
( S i ,k − vi m k ) 2
i =1
vi
∑
The formula means that we “downweight” the squared residuals for those
observations with large variances. It turns out that the solution to (5.3) is
23
indeed the estimate m̂ k from equation (2.1), and the variance of this
estimator coincides with the variance given by (2.4)! Thus the whole
machinery known from statistical textbooks to test the linearity (AM 1) and the
variance assumption (AM 3) can be applied. This is usually done by plotting
the data and residuals.
As a start we may plot the vi against the S i , k in order to confirm that there is
indeed a reasonable linear relationship.
As a next step we plot the residuals, i.e. the deviations of the known
outcomes to the fitted curve. In general, if the chosen model is a good one,
then the residuals should reflect only random fluctuation of the data around
the fitted line. To be precise, we plot the weighted residuals, since we are
dealing with a weighted regression where the deviations from the fitted curve
are expected to “fan out”.
Thus, we plot the residuals
(5.4)
S i , k − vi mˆ k
vi
against the exposure vi . Ideally, this graph will show no pattern but only
“noise”.
In practice, it is hard to draw conclusions from this check for small triangles or
development years with thin data, i.e. the later development years.
5.2 Testing the Multiplicative Model
The Chain-Ladder model can be tested in the same way. For the sake of
completeness we repeat the basic findings. First, let us recall the three basic
Chain-Ladder assumptions:
(CL 1) E (Ci ,k Ci1 ,K, Cik −1 ) = Ci ,k −1 f k
(CL 2) Independence of the underwriting years
(CL 3) Var (Ci ,k Ci1 ,K, Cik −1 ) = Ci ,k −1σ k2
Again, for a fixed k, the model-assumptions can be viewed as a
heteroskedastic linear regression.
The best linear unbiased estimator for f k is determined via weighted least
squares by minimising the expression
(5.5)
n +1− k
(Ci ,k − Ci ,k −1 f k ) 2
i =1
Ci ,k −1
∑
24
and its unique solution is the Chain-Ladder estimate fˆk from equation (3.5),
and its variance coincides with the variance given by (3.8)!
We may plot the C i , k −1 against the C i , k in order to confirm that there is indeed
a reasonable linear relationship. In practice, it might be better to plot
S i ,k = C i ,k − C i ,k −1 against C i , k instead, the required slope of the regression line
is then f k − 1 .
The corresponding residuals become
(5.6)
C i ,k − C i ,k −1 fˆk
C i ,k −1
=
S i ,k − C i ,k −1 ( fˆk − 1)
C i ,k −1
which are plotted against the known C i , k −1 . This graph should show no pattern
but only “noise”.
It is worth noting that the residual analysis can be further refined to test if
other variance assumptions than (CL 3) are more in line with the given data.
A thorough discussion is beyond of the scope of this article. The interested
reader is referred to Mack [1994].
5.3 Checking for calendar-year effects
Before we could to apply the regression-analysis in the preceding subsection,
we had to assume that (CL 2) is already fulfilled (or that at least the
underwriting years are uncorrelated). As mentioned before, this is in fact a
quite strong statement because in reality there is a range of factors which may
violate such an assumption. It is out of the scope of this article to introduce a
thorough test for independence. The reader who is seeking more depth is
referred to Mack [1994]. Here, we illustrate a simple but useful test to check
for so-called calendar-year effects. And indeed, one of the major reasons
why the independence of the underwriting years (CL 2) is violated can be
attributed to calendar-year effects. Calendar-year effects may be caused by
changes of legislation or claims handling or in irregular inflation, to name but a
few.
Calendar-year effects are acting on the diagonals of the run-off triangle, i.e.
they affect several underwriting years in different development periods.
Statistically speaking, a calendar-year effect will produce increment ratios or
cumulative losses that are systematically higher (or lower) than what would
have been predicted by the increment ratios or development factors by the
model.
Thus, if we plot the standardised residuals
25
(5.8)
S i ,k − vi mˆ k
sˆk vi
1≤ i + k ≤ n
or
(5.7)
Ci ,k − C i ,k −1 fˆk
σˆ k C i ,k −1
1≤ i + k ≤ n ,
against the corresponding calendar-year i+k-1 we expect only “white noise” if
the model-assumptions are fulfilled. If, on the other hand, these residuals all
have the same sign for a fixed calendar-year, then this could be an indication
for a calendar-year effect and must be further investigated since a mechanical
application of the Incremental Loss Ratio or Chain-Ladder method would now
use increment ratios that are systematically too high in every development
period! It is probably safest to disregard the whole diagonal by means of
setting “weights” as will be explained in the next section.
Apart from spotting calendar-year effects, plotting the residuals (5.7) and (5.8)
can also help detecting large deviations of single individual development
factors from their weighted mean. The graph then becomes a useful tool in
connection with the setting individual weights and can be quickly analysed. A
more formal statistical test for calendar-year effects is developed in Mack
[1994].
26
SECTION 6: PRACTICAL CONSIDERATIONS
6.1 Dealing with outliers
In almost cases in practice, it will not make sense to apply the Mack too
mechanically, because the data may be distorted by outliers and calendaryear effects – or one may feel that it is more appropriate to use only the last
few diagonals because the underlying business has changed, to name but a
few reasons. Consequently, one may decide to manually adjust the
respective development factors and in principle there is nothing wrong with
that approach. However, this procedure leaves the actuary with the problem
of also adjusting the corresponding sigmas or standard errors, which is less
intuitive. Yet there is a more precise way of how to tackle this issue. We only
need to “rewrite” all formulas, this time omitting those individual increment
ratios or development factors which we regard as outliers. The formal
procedure and underlying theory are in fact quite simple, however it seems
that it has not been fully appreciated by the actuarial literature. An example of
how to realise the proposed procedure in practice can be found in the
accompanying Excel spreadsheets.
Formalism of setting weights
Formally, the approach looks as follows: We start by looking again at the
formula for the incremental loss ratio, expressed as a weighted mean. In
principle there is nothing stopping us from using a different set of weights, say
wi , k ⋅ S i ,k in order to reflect the credibility we put into S i ,k . In this case, the
corresponding formulas for the increment ratios translate naturally to
n +1− k
(6.1) mˆ k =
n +1− k
∑
i =1
∑ (w
⎛
⎞
wi ,k ⋅ v i
⎜
⎟=
ˆ
m
⎜ n +1− k ( w v ) ik ⎟
i ,k i
⎝ ∑i =1
⎠
i =1
n +1− k
i ,k
∑ (w
i =1
i ,k
⋅ S ik )
⋅ vi )
and for the parameters relating to the s.e. we have
⎞
⎛S
1 n +1− k
(6.2) sˆ =
wi , k ⋅ vi ⎜⎜ ik − mˆ k ⎟⎟
∑
| I | −1 i =1
⎠
⎝ vi
2
2
k
where | I | denotes the number of incremental loss ratios that are taken into
account in year k and
(6.3) Var (mˆ k ) =
∑
s k2
n +1− k
i =1
( wi ,k ⋅ vi )
27
From here we can proceed in the usual way, say via the recursion formula
etc.
The modification through weights corresponds to slightly modified versions of
the original model assumptions. A more rigorous discussion for the ChainLadder method can be found in Mack [1999]. As long as we are only
“deselecting” outliers we are only interested in the case where wi ,k ∈ {0,1} and
the modifications stay simple.
In a similar fashion the formulas in the Chain-Ladder method can be amended
to account for outliers. The formulas for the development factors and the
sigmas become then, respectively:
n +1− k
(6.4) fˆk =
n +1− k
∑
i =1
⎛
⎞
wi ,k ⋅ C i ,k −1
⎜
ˆ
⋅F ⎟ =
⎜ n +1− k ( w C ) i ,k ⎟
i , k i , k −1
⎝ ∑i =1
⎠
∑ (w
⋅ C i ,k )
∑ (w
⋅ C i ,k −1 )
i =1
n +1− k
i ,k
i =1
⎛ Ci ,k
⎞
1 n +1− k
(6.5) σˆ =
wi ,k C i ,k −1 ⎜⎜
− fˆk ⎟⎟
∑
| I | −1 i =1
⎝ C i ,k −1
⎠
i ,k
2
2
k
where | I | denotes the number of individual development factors that are
taken into account and
( )
(6.6) Var fˆk =
∑
σ k2
n +1− k
i =1
( wi ,k ⋅ C i ,k −1 )
The weighting procedure as introduced herein may look harmless and it is
indeed mathematically trivial. Yet it is probably one of the key mechanisms to
be employed when working in practice and can also be incorporated when
working with spreadsheets as the accompanying workbook shall demonstrate.
6.2 Deciding between the multiplicative and additive model
General remarks
So far we have introduced two different methods of projecting claims, and
have given some basic statistical tests to test the underlying model
assumptions. Apart from the statistical tests, it is also important to have a
less formal understanding of how these projection-methods work and where
the caveats lie. The reality will often be that one has to decide which model is
the “lesser evil”.
In an imprecise way we may first note the following: if the outstanding reserve
is mainly determined by the run-off of already reported claims, this is an
indication that the multiplicative method may lead to more stable results. If on
28
the other hand the portfolio is more affected by genuinely late-reported IBNRclaims, the additive method might be favoured, as the additive method is
indifferent to the loss development that has been observed so far.
Furthermore, for the most recent underwriting years (the data in the lower left
corner of the triangle) the multiplicative method may not applicable because
no losses have been reported, and thus the multiplicative projection will
forecast 0 claims. In this case one might be forced either to apply some
additive projection for the most recent underwriting years, or estimate the lost
in the first development year by other means.
Another important remark must be made with respect to the Incremental Loss
Ratio method: if the underwriting-year premium is chosen as an
exposure measure, then the Incremental Loss-ratio Method makes only
sense if the premium-volume does in fact reflect the size of the
underlying business. In other words, premium cycles or rate hardenings
need to be corrected before the additive method can be applied.
Overall, the choice of model and the degree of trust that the actuary may put
into their results will always be of judgement and experience.
Graphical interpretation
Another very simple yet quite useful tool in analysing the appropriateness of
the respective model-assumptions consists of plotting appropriate graphs of
the run-off data.
The rational behind this procedure is nothing but the fundamental assumption
that the future payments will be somehow similar to what has happened in the
past. This is so obvious that it seems hardly worth mentioning. Yet this
remark leads to a very simplistic but useful rule of thumb when the actuary
has to decide which of the two proposed methods could be given preference.
The key assumption in the additive case is that the loss ratios increase by a
certain amount in a particular development year, the amount being the same
for all underwriting years, except for random noise. In formulas:
(6.7)
Ci ,k
vi
=
Ci ,k −1
vi
+ mk
Therefore, a plot of the development periods against loss ratios for all
underwriting years should show that the resulting graphs are parallel to each
other.
In the multiplicative case the key assumption was that within a specified
development year, the cumulative loss amounts of each underwriting year are
multiplied by the same factor except for random noise. This in turn means
29
that the logarithms of the cumulative losses increase by the same amount.
Speaking in formulas, we have for the Chain-Ladder model:
(6.8) log C i ,k = log C i , k −1 + log f k
As a result, the graphs of the underwriting-years in a dev-year/logarithmic
cumulative-loss-space should be parallel.
To sum it up, a rough check as to wether the data has behaved according to
either model consists of plotting two graphs looking roughly like the following:
Loss ratios
1990
250%
1991
1992
200%
1993
150%
100%
Losses in $'000, log-scale
10,000
1992
1993
1994
1995
1995
1996
1997
1996
1997
1994
1,000
1998
1999
50%
1998
1999
2000
2001
2000
2001
2002
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Development years
1990
1991
2003
2004
2002
100
1
2
3
4
5
6 7 8 9 10 11 12 13 14 15
Development years
2003
2004
If we intend to apply the Incremental Loss Ratio method, then the graph on
the left hand side should show a parallel pattern. And if we intend to apply the
Chain-Ladder method, then the graph on the right hand side is the appropriate
interpretation of the data.
In fact, the additive or multiplicative projection methods can be interpreted as
“parallelly projecting” the loss experience of the older underwriting years in a
loss ratio graph or logarithmic dollar-graph, respectively. The incremental
loss ratio or development factor in year k is nothing but an average of the
observed slopes in that development year.
The graphical representations are often helpful in many respects, not only
when deciding between the two models. It may be possible to spot trends,
outliers or other peculiarities. For example, in the above data-example, we
can spot immediately an erratic spike in years 2 and 3 of the underwriting year
1997. As a consequence, these two individual development factors must be
taken out by means of the weighting procedure described above.
Furthermore, it is sometimes possible to immediately identify an appropriate
range of underwriting periods and calendar years in which the data show the
most homogenous pattern and just “get a feel” for the run-off behaviour.
Overall, the usefulness of these two graphs when interpreting either method
can hardly be overestimated.
30
6.3 Reduction of parameters
As a final refinement we discuss how the sigmas and development factors
might be smoothened or the run-off extended via regression techniques. This
section is based on Mack [1997, 2002]. The reader who is not familiar with
weighted regressions may find this section difficult, although the basic idea of
fitting a curve through the parameters is quite straightforward.
The Sigma-Regression
We have already indicated that the “sigmas” of both methods can be quite
volatile without indicating how to possibly deal with this difficulty.
The considerations in this subsection can in principle be applied to both the
additive and the multiplicative method. However, it seems that in the additive
case it is much harder to fit a curve through the sˆk2 than through the σˆ k2 ,
which is why we restrict our discussion to the Chain-Ladder method. Yet it
must be pointed out that for both methods there is no logical connection
leading from the model assumptions to the smoothening-approach suggested
here.
As a starting point, one might first want to plot the σˆ k2 on a log-linear scale
and investigate any extreme outliers by analysing again the triangle of
individual development factors. If the σˆ k2 seem to decline in a log-linear
fashion from a certain development period k onwards, a formal curve fitting
approach becomes:
(6.9) Var (C i ,k C i1 ,K , C ik −1 ) = C i ,k −1σ 2 e − ck
This approach does not affect the optimal estimators for f k since the σˆ k2 differ
from each other only through a constant factor. In theory, the “most suitable”
regression for c and σ 2 depends on the respective variances of the σˆ k2 which are unknown. Yet some intuitive reasoning can help: it is plausible to
expect the respective unknown variances of the σˆ k2 to decrease, the more
data points are included in their estimation, with the decrease proportional to
the number of data points taken into account (e.g. the variance of the
estimator of a sample mean shows exactly this behaviour). Therefore it
seems at least plausible to use the reciprocal of (n-k-1) as a weight for σˆ k2 in a
regression. The corresponding expression to be minimised in order to
estimate σ and c can then be written as
∑ (n − k − 1)(log(σˆ
n−2
(6.10)
k =1
2
k
) − log(σ 2 e −ck )
)
2
There is no neat expression for the solutions for σ and c, and some statistical
computer package or a solver function might be the method of choice.
31
The resulting smoothed sigmas may then be used to recalculate the standard
errors of the factors and estimators, in particular for the first and last
development years.
Regression of development factors
In a similar way, we can fit a curve thorough the development factors in order
to smoothen or extend the run-off. Again, experience shows that in the
multiplicative case the factors often decline in a log-linear fashion, at least
from a certain development year on (e.g. from the 2nd or 3rd development
year). So we may again start with a relation like
(6.11) f k = 1 + ae − bk
and then determine a and b. As with the sigmas, we should weight the
regression with the variances of the f k , which we have already calculated at
this stage.
Therefore, the expression to be minimised becomes
( fˆk − 1 − ae − bk ) 2
∑
Var ( fˆ )
k =2
n −1
(6.12)
k
In order to keep the expressions as simple as possible, we may use the σˆ k2
from the sigma-regression, in which case minimising (6.12) is equivalent to
minimising
n −1
(6.13)
∑
k =2
( fˆk − 1 − ae − bk ) 2
∑
n +1− k
i =1
Ci ,k
Again, there is no neat explicit expression to solve (6.13) but some solverfunction will do the trick.
The biggest difficulty that arises from here consists of the derivation of a
theoretically correct standard error that corresponds to the factors from the
regression. This is far from trivial and may depend on additional assumptions.
We will not address this difficult issue in this paper. One may use formulas
like (3.8) together with the sigma-regression, and then apply the recursionformulae, although as a word of warning it must be stressed that this
approach is not easily justified theoretically.
32
Appendix A: Formulas for the Incremental Loss Ratio method
The increment ratio
n +1 − k
mˆ k =
∑S
i =1
n +1 − k
ik
∑v
i
i =1
Projected loss in underwriting year i and development year k+1:
Cˆ i ,k +1 = Cˆ i ,k + vi ⋅ mˆ k
Sigma:
⎞
⎛S
vi ⎜⎜ ik − mˆ k ⎟⎟
⎠
⎝ vi
n +1− k
1
sˆ =
n−k
∑
2
k
i =1
2
Standard error of the incremental Loss Ratio:
s.e.(mˆ k ) =
2
∑
sˆk2
n +1− k
i =1
vi
Standard error for the projected loss underwriting year i and
development year k:
(
)
2
s.e. Cˆ i ,k +1
( )
2
= vi ⋅ sˆk2 + vi2 ⋅ s.e.(mˆ k ) + s.e. Cˆ i ,k
The standard error of the total reserve:
u k = ∑i = n − k + 2 v i
n
( )
s.e. Cˆ k +1
2
( )
2
= u k ⋅ sˆk2 + u k2 ⋅ s.e.(mˆ k ) + s.e. Cˆ k
2
33
Appendix B: Formulas for the Chain-Ladder method
The development factor:
n +1− k
fˆk =
∑C
i =1
n +1− k
∑C
i =1
i ,k
i , k −1
Projected loss:
Cˆ i ,k = fˆk ⋅ Cˆ i ,k −1
Sigma:
⎛ Ci ,k
⎞
1 n +1− k
− fˆk ⎟⎟
σ̂ =
C i ,k −1 ⎜⎜
∑
n − k i =1
⎝ C i ,k −1
⎠
2
2
k
Standard error of the development factor:
( )
s.e. fˆk
2
=
∑
σˆ k2
n +1− k
i =1
C i ,k −1
Standard error for the projected loss underwriting year i and
development year k:
(
)
2
s.e. Cˆ i ,k +1
( )
= Cˆ i ,k σˆ k2+1 + Cˆ i2,k s.e.( fˆk +1 ) 2 + fˆk2+1 ⋅ s.e. Cˆ i ,k
The standard error of the total reserve:
hk = ∑i = n − k +1 C i ,k
n
( )
s.e. Cˆ k +1
2
( )
= hk σˆ k2+1 + hk2 ⋅ s.e.( fˆk +1 ) 2 + fˆk2+1 ⋅ s.e. Cˆ k
34
2
2
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35